
Ricci Tensor
\(R_{\mu \nu }= \Big [\dot{X}+H\Big ( (n-1)X-(n-2)Y+(Z+V)\Big ) -(n-2)X Y+X(Z+V)-2 W^{2}+\dot{H}+(n-1)H^{2}\Big ] h_{\mu \nu }\nonumber \\+\Big [ \dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big ](n-1)u_{\mu }u_{\nu }\) 

Homothetic Curvature
\(\hat{R}_{\mu \nu } \equiv 0\) 

Co-Ricci
\(\breve{R}_{\mu \nu }=\Big [ \dot{Y}+H \Big (-(n-2)X+(n-1)Y+(Z+V) \Big )+(n-2)X Y -Y (Z+V) + 2 W^{2} -\dot{H}-(n-1)\dot{H}^{2}\Big ]h_{\mu \nu } \nonumber \\+\Big [ \dot{X}+H (X+Z+V)+X (Z+V)+(\dot{H}+H^{2}) \Big ] (n-1)u_{\mu }u_{\nu }\) 

Ricci Scalar
\(R=(n-1)\Big [(\dot{X}-\dot{Y})+(n-1) H (X-Y) +(X+Y)(Z+V)-(n-2) X Y - 2 W^{2}+2 \dot{H}+n H^{2} \Big ]\) 

Ricci+co-Ricci
\((R_{\mu \nu }+ \breve{R}_{\mu \nu })=\Big [ (\dot{X}+\dot{Y})+H (X+Y)+2H (Z+V) +(X-Y)(Z+V) \Big ]\Big (h_{\mu \nu }+(n-1)u_{\mu }u_{\nu }\Big )\) 

Ricci-co-Ricci
\((R_{\mu \nu }- \breve{R}_{\mu \nu })=\Big [ (\dot{X}-\dot{Y})+(2n-3) H (X-Y) +(X+Y)(Z+V) -2(n-2) X Y -4 W^{2}+2 \dot{H}+2 (n-1)H^{2}\Big ]h_{\mu \nu }\nonumber \\-\Big [ (\dot{X}-\dot{Y})+ H (X-Y)+ (X+Y)(Z+V)+2 (\dot{H}+H^{2})\Big ](n-1)u_{\mu }u_{\nu }\) 

Generalized Einstein TensorThe definition we adopt for the Einstein tensor comes from the contraction of the generalized Riemann double dual. Our definition agrees with the one used in [1]}. We should note however that this generalized version has neither properties of the usual Einstein tensor, that is the generalized Einstein tensor is, in general, neither symmetric nor covariantly conserved.
\(G_{\mu \nu }:=\frac{(n-3)}{2}\Big ( R_{\mu \nu }-\breve{R}_{\mu \nu }-R g_{\mu \nu }\Big )= \nonumber \\=\frac{(n-2)(n-3)}{2}\Big [ -\Big ( (\dot{X}-\dot{Y})+(n-2)H(X-Y)-(X+Y)(Z+V)+(n-3)X Y-W^{2}+2 \dot{H}-H^{2} \Big ) h_{\mu \nu }\nonumber \\+\Big ( H(X-Y)-X Y-W^{2}+H^{2}\Big )(n-1)u_{\mu }u_{\nu } \Big ]\) 

Right dual
\(R_{\mu \nu \alpha \beta }\varepsilon ^{\alpha \beta \kappa \lambda } =\Big [ \dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big ] 2 u_{\mu }\varepsilon _{\nu }^{\;\;\kappa \lambda }\nonumber \\+\Big [ \dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big ]2 u_{\nu }\varepsilon _{\mu }^{\;\;\kappa \lambda }\nonumber \\+\Big [H^{2}+ H(X-Y)-X Y-W^{2}\Big ] 2 \Big ( \varepsilon _{\mu \nu }^{\;\;\;\;\kappa \lambda }+u_{\nu }\varepsilon _{\mu }^{\;\;\kappa \lambda }-u_{\mu }\varepsilon _{\nu }^{\;\;\kappa \lambda }\Big )+8 XW u_{\mu }\delta _{\nu }^{[\kappa }u^{\lambda ]}+8 Y W u_{\nu }\delta _{\mu }^{[\kappa }u^{\lambda ]}\nonumber \\+4(\dot{W}-H W)h^{\kappa }_{\;\;[\mu }h_{\nu ]}^{\;\;\;\lambda }+8 HW \delta ^{[\kappa }_{\mu }\delta ^{\lambda ]}_{\nu }\) 

Symmetric Part of Riemann
\(R_{(\mu \nu )\alpha \beta }=\Big [ (\dot{X}+\dot{Y})+H(X+Y)+2 H(Z+V)+(X-Y)(Z+V)\Big ] 2 u_{(\mu }h_{\nu )[\alpha }u_{\beta ]}-2 W(X+Y)\varepsilon _{\alpha \beta (\mu }u_{\nu )}\) 

Totally Antisymmetric RiemannIn the literature this term is sometimes called the Hojman or more frequently the Holst term.
\(\varepsilon ^{\mu \nu \alpha \beta }R_{\mu \nu \alpha \beta }=12\Big [ (\dot{W}+3 H W)+W ( X-Y) \Big ]\) 

